Federated learning has gained popularity as a means of training models distributed across the wireless edge. The paper introduces delay-aware hierarchical federated learning (DFL) to improve the efficiency of distributed machine learning (ML) model training by accounting for communication delays between edge and cloud. Different from traditional federated learning, DFL leverages multiple stochastic gradient descent iterations on local datasets within each global aggregation period and intermittently aggregates model parameters through edge servers in local subnetworks. During global synchronization, the cloud server consolidates local models with the outdated global model using a local-global combiner, thus preserving crucial elements of both, enhancing learning efficiency under the presence of delay. A set of conditions is obtained to achieve the sub-linear convergence rate of O(1/k) for strongly convex and smooth loss functions. Based on these findings, an adaptive control algorithm is developed for DFL, implementing policies to mitigate energy consumption and communication latency while aiming for sublinear convergence. Numerical evaluations show DFL's superior performance in terms of faster global model convergence, reduced resource consumption, and robustness against communication delays compared to existing FL algorithms. In summary, this proposed method offers improved efficiency and results when dealing with both convex and non-convex loss functions.

This paper provides a finite-sample analysis of a passive stochastic gradient Langevin dynamics algorithm (PSGLD) designed to achieve adaptive inverse reinforcement learning (IRL). By passive, we mean that the noisy gradients available to the PSGLD algorithm (inverse learning process) are evaluated at randomly chosen points by an external stochastic gradient algorithm (forward learner) that aims to optimize a cost function. The PSGLD algorithm acts as a randomized sampler to achieve adaptive IRL by reconstructing this cost function nonparametrically from the stationary measure of a Langevin diffusion. Previous work has analyzed the asymptotic performance of this passive algorithm using weak convergence techniques. This paper analyzes the non-asymptotic (finite-sample) performance using a logarithmic-Sobolev inequality and the Otto-Villani Theorem. We obtain finite-sample bounds on the 2-Wasserstein distance between the estimates generated by the PSGLD algorithm and the cost function. Apart from achieving finite-sample guarantees for adaptive IRL, this work extends a line of research in analysis of passive stochastic gradient algorithms to the finite-sample regime for Langevin dynamics.

Stochastic optimization methods such as mirror descent have wide applications due to low computational cost. Those methods have been well studied under assumption of the independent and identical distribution, and usually achieve sublinear rate of convergence. However, this assumption may be too strong and unpractical in real application scenarios. Recent researches investigate stochastic gradient descent when instances are sampled from a Markov chain. Unfortunately, few results are known for stochastic mirror descent. In the paper, we propose a new version of stochastic mirror descent termed by MarchOn in the scenario of the federated learning. Given a distributed network, the model iteratively travels from a node to one of its neighbours randomly. Furthermore, we propose a new framework to analyze MarchOn, which yields best rates of convergence for convex, strongly convex, and non-convex loss. Finally, we conduct empirical studies to evaluate the convergence of MarchOn, and validate theoretical results.

In this paper, we present an adaptation of Newton's method for the optimization of Subspace Support Vector Data Description (S-SVDD). The objective of S-SVDD is to map the original data to a subspace optimized for one-class classification, and the iterative optimization process of data mapping and description in S-SVDD relies on gradient descent. However, gradient descent only utilizes first-order information, which may lead to suboptimal results. To address this limitation, we leverage Newton's method to enhance data mapping and data description for an improved optimization of subspace learning-based one-class classification. By incorporating this auxiliary information, Newton's method offers a more efficient strategy for subspace learning in one-class classification as compared to gradient-based optimization. The paper discusses the limitations of gradient descent and the advantages of using Newton's method in subspace learning for one-class classification tasks. We provide both linear and nonlinear formulations of Newton's method-based optimization for S-SVDD. In our experiments, we explored both the minimization and maximization strategies of the objective. The results demonstrate that the proposed optimization strategy outperforms the gradient-based S-SVDD in most cases.

Federated learning has emerged as a viable distributed solution to train machine learning models without the actual need to share data with the central aggregator. However, standard neural network-based federated learning models have been shown to be susceptible to data leakage from the gradients shared with the server. In this work, we introduce federated learning with variational quantum circuit model built using expressive encoding maps coupled with overparameterized ans\"atze. We show that expressive maps lead to inherent privacy against gradient inversion attacks, while overparameterization ensures model trainability. Our privacy framework centers on the complexity of solving the system of high-degree multivariate Chebyshev polynomials generated by the gradients of quantum circuit. We present compelling arguments highlighting the inherent difficulty in solving these equations, both in exact and approximate scenarios. Additionally, we delve into machine learning-based attack strategies and establish a direct connection between overparameterization in the original federated learning model and underparameterization in the attack model. Furthermore, we provide numerical scaling arguments showcasing that underparameterization of the expressive map in the attack model leads to the loss landscape being swamped with exponentially many spurious local minima points, thus making it extremely hard to realize a successful attack. This provides a strong claim, for the first time, that the nature of quantum machine learning models inherently helps prevent data leakage in federated learning.

In this study, we propose a digital over-the-air computation (OAC) scheme for achieving continuous-valued (analog) aggregation for federated edge learning (FEEL). We show that the average of a set of real-valued parameters can be calculated approximately by using the average of the corresponding numerals, where the numerals are obtained based on a balanced number system. By exploiting this key property, the proposed scheme encodes the local stochastic gradients into a set of numerals. Next, it determines the positions of the activated orthogonal frequency division multiplexing (OFDM) subcarriers by using the values of the numerals. To eliminate the need for precise sample-level time synchronization, channel estimation overhead, and channel inversion, the proposed scheme also uses a non-coherent receiver at the edge server (ES) and does not utilize a pre-equalization at the edge devices (EDs). We theoretically analyze the MSE performance of the proposed scheme and the convergence rate for a non-convex loss function. To improve the test accuracy of FEEL with the proposed scheme, we introduce the concept of adaptive absolute maximum (AAM). Our numerical results show that when the proposed scheme is used with AAM for FEEL, the test accuracy can reach up to 98% for heterogeneous data distribution.

We present Zeroth-order Riemannian Averaging Stochastic Approximation (\texttt{Zo-RASA}) algorithms for stochastic optimization on Riemannian manifolds. We show that \texttt{Zo-RASA} achieves optimal sample complexities for generating $\epsilon$-approximation first-order stationary solutions using only one-sample or constant-order batches in each iteration. Our approach employs Riemannian moving-average stochastic gradient estimators, and a novel Riemannian-Lyapunov analysis technique for convergence analysis. We improve the algorithm's practicality by using retractions and vector transport, instead of exponential mappings and parallel transports, thereby reducing per-iteration complexity. Additionally, we introduce a novel geometric condition, satisfied by manifolds with bounded second fundamental form, which enables new error bounds for approximating parallel transport with vector transport.

Abstract--One of the crucial issues in federated learning is how to develop efficient optimization algorithms. Most of the current ones require full device participation and/or impose strong assumptions for convergence. Different from the widely-used gradient descentbased algorithms, in this paper, we develop an inexact alternating direction method of multipliers (ADMM), which is both computationand communication-efficient, capable of combating the stragglers' effect, and convergent under mild conditions. Furthermore, it has a high numerical performance compared with several state-of-the-art algorithms for federated learning. This idea has been extensively exploited in the [4], [5], [6], digital health [7], and mobile edge and over-theair stochastic gradient descent (SGD) algorithms, such as the computing [8], [9], [10], [11].

In a recent paper, Ling et al. investigated the over-parametrized Deep Equilibrium Model (DEQ) with ReLU activation. They proved that the gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. This paper shows that this fact still holds for DEQs with any general activation that has bounded first and second derivatives. Since the new activation function is generally non-linear, bounding the least eigenvalue of the Gram matrix of the equilibrium point is particularly challenging. To accomplish this task, we need to create a novel population Gram matrix and develop a new form of dual activation with Hermite polynomial expansion.

Langevin Monte Carlo (LMC) and its stochastic gradient versions are powerful algorithms for sampling from complex high-dimensional distributions. To sample from a distribution with density $\pi(\theta)\propto \exp(-U(\theta)) $, LMC iteratively generates the next sample by taking a step in the gradient direction $\nabla U$ with added Gaussian perturbations. Expectations w.r.t. the target distribution $\pi$ are estimated by averaging over LMC samples. In ordinary Monte Carlo, it is well known that the estimation error can be substantially reduced by replacing independent random samples by quasi-random samples like low-discrepancy sequences. In this work, we show that the estimation error of LMC can also be reduced by using quasi-random samples. Specifically, we propose to use completely uniformly distributed (CUD) sequences with certain low-discrepancy property to generate the Gaussian perturbations. Under smoothness and convexity conditions, we prove that LMC with a low-discrepancy CUD sequence achieves smaller error than standard LMC. The theoretical analysis is supported by compelling numerical experiments, which demonstrate the effectiveness of our approach.